Definition 6 ([1]}) 
Let the profile of \(M\)  be given by the vector \(p(M) = \langle p_1\dots ,p_L \rangle \)  where \(p_k = | \lbrace  a_i\in A^* : \operatorname{rank}(a_i,M(a_i))=k \rbrace | \)  and \(L\)  is the maximum length of any agent's preference list. This means that the profile describes the number of first, second, etc. choices in a matching.

